Module structure of the Lie algebra $W_n(K)$ over $s{l}_n(K)$

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, $A=\mathbb{K}[x_1,\dots ,x_n]$ the polynomial ring, and let $W_n(\mathbb{K})$ denote the Lie algebra of all $\mathbb{K}$-derivations on $A$. The Lie algebra $W_n:=W_n(\mathbb{K})$ admits a natural grading $W_n={⨁}_{i\ge -1}{W}_n^{[i]}$, where $W_n^{[i]}$ consists of all homogeneous derivations whose coefficients are homogeneous polynomials of degree $i+1$ or zero. The component $W_n^{[0]}$ is a subalgebra of $W_n$ and is isomorphic to ${\mathfrak{gl}}_n(\mathbb{K}).$ Moreover, each $W_n^{[i]}$ for $i\ge -1$ is a finite-dimensional module over $W_n^{[0]}$. We prove that $W_n^{[i]},i\ge 0$ is a sum of two irreducible submodules $W_n^{[i]}=M_i\oplus N_i$, where $M_i$ consists of all divergence-free derivations, and $N_i$ consists of derivations that are polynomial multiples of the Euler derivation $E_n={\sum }_{i=1}^n{x}_i\frac{\partial }{\partial x_i}$. As a consequence, we show that the standard grading is exact in certain sense, namely: $[W_n^{[i]},W_n^{[j]}]=W_n^{[i+j]}$ for all $i,j,$ except when $i=j=0$. We also address the question of when the subalgebra of $W_n$ generated by $W_n^{[-1]}\oplus W_n^{[0]},$ together with an additional element from $W_n,$ equals the entire Lie algebra $W_n$.